What is the standard deviation/uncertainty of a single observation in a meta-analysis?

Question

I am running a simulation with parameters based on an existing random-effects meta-analysis. I would like to draw individual patients in the treatment group from a normal distribution with µ=the overall estimated treatment effect from the random-effects model, but I am struggling to come up with the standard deviation.

I know that I cannot just use the standard error of the random-effects model, since this only measures the uncertainty of the overall estimated treatment effect. I need to somehow take into account the standard deviations in the original studies, as well as the between-study heterogeneity/variance of true effect sizes τ², but am not sure how to connect the two. I would appreciate any ideas.